978-0393919684 Chapter 17 Lecture Note

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CHAPTER 17

Bargaining

Teaching Suggestions

Newspapers often print articles of advice on bargaining for buyers of houses, autos,

and so on. We found a particularly good one that we mention in the text: Andrée Brooks,

“Honing Haggling Skills” (New York Times, December 5, 1993). It mentions (not in technical

language) several points that relate to the concepts in this chapter: best alternative to a negoti –

ated agreement (BATNA), patience, signaling, screening, using mandated agents for credible

commitment, and how multiple dimensions of bargaining facilitate agreement by exploiting

differences in the parties’ relative valuations of the dimensions. You can circulate such an

article to your class a day in advance and start with a discussion that elicits these concepts and

builds them into the game-theoretic framework of the chapter.

You can also look for other popular press publications or Web sites and interpret their

content using the bargaining terminology found in this chapter. Consider the book Getting to

Yes by R. Fisher and W. Ury (Baltimore, Md.: Penguin, 1981) or an Internet site that addresses

bargaining in the process of purchasing a car:

www.edmunds.com/car-buying/negotiating-101.html.

When you cover the alternating-offer models of bargaining, you may find it easier for

your students to follow the progression of the game if they have a simple table showing the

offers and amounts going to each player in each round. If there is gradual decay in the total

value available, we say that the total value drops by x1 after the first offer is rejected and

further rejections lead to drops of x2, x3, . . . (all drops are the same size). For a game where $1

is being split and the value drops by 10 cents after each rejected offer, the last possible offer

would occur when only x10 was left. This bargaining situation can be illustrated with the

following table:

Round Offer by A gets B gets

10 B 0 x10

9 A x9x10

8 B x9x8 + x10

7 A x7 + x9x8 + x10

6 B x7 + x9x6 + x8 + x10

1 A x1 + x3 + x5 + x7 + x9x2 + x4 + x6 + x8 + x10

The final row of the table shows the division that can be anticipated as the Nash equilibrium of

the bargaining game.

For a bargaining game with impatience, let A regard $1 immediately as equivalent to

$(1 + r) > $1 one offer later and B regard $1 immediately as equivalent to $(1 + s) > $1 one

offer later. The values r and s measure impatience. Let a = 1/(1 + r) and b = 1/(1 + s). Then in a

game of dividing $1 that could go, say, 100 rounds, the table of offers and payoffs is as follows:

Round Offer by A gets B gets

100 B 0 1

99 A 1 – b b

98 B a(1 – b) 1 – a + ab

97 A 1 – b + ab – ab2b(1 – a + ab)

96 B a(1 – b + ab – ab2) 1 – a + ab – a2b + a2b2

Whenever A makes an offer, she will ask for

x = 1 – b + ab – ab2 + a2b2 – a2b3 + . . .

= (1 – b)(1 + ab + a2b2 + a3b3 + . . .).

The limit, with an indefinite repetition of offers, yields

x = (1 – b)/(1 – ab)

= (s + rs)/(r + s + rs).

Similarly, whenever B makes an offer, she will ask for

y = (r + rs)/(r + s + rs).

Game Playing in Class

You have probably already played some simple bargaining games in the very first

session or two of your class—the take-it-or-leave-it-offer game, the Centipede game, and so

on. Now you can simply remind the class or play some of the games again, compare their

behavior after learning game theory with their behavior before, and discuss this as a matter of

interest in its own right. We set out some options for in-class bargaining over $10 below. You

may also want to try some of the online bargaining games available through Charlie Holt’s

Veconlab site (veconlab.econ.virginia.edu/admin.html).

GAME 1—Bargaining over $10

This is a collection of short bargaining situations that students can be asked to consider

for homework or in class. Variants on these situations are easily created. Instructions: The

minimum offer in all of these situations is 50 cents. Each of you will be paired against the

instructor when determining points. The instructor will respond to your offers using the Nash

equilibrium strategy. If your offer is declined in any of the games, it will be assumed that the

Nash equilibrium outcome is obtained at the next offer.

Situation 1

You and a partner are given $10 to split between you. You make the first move and offer your

partner part of the $10. If she accepts, the bargain is complete and you each get an amount

determined by your offer. If she declines, she gets a turn and makes you an offer. Again, the

bargain is complete if you accept, and the split depends on her offer. This process continues,

infinitely if necessary, until one of you makes an offer that the other accepts.

If you are making the first offer in this bargaining game, what do you offer?

_________________

Situation 2

You and a partner are given $10 to split between you. You will make the first move and offer

your partner a portion of the $10. If she accepts, the bargain is complete and you each get an

amount determined by your offer. If she declines, the $10 shrinks to $5 and she then gets a turn

to make you an offer. Again, the bargain is complete if you accept and the division is made

according to the terms of her offer. If you decline her offer, the $5 shrinks to zero, the game

ends, and both of you receive nothing.

If you are making the first offer in this bargaining game, what do you offer?

_________________

Situation 3

You and a partner are given $10 to split between you. You will make the first move and offer

your partner a portion of the $10. If she accepts, the bargain is complete and you each get an

amount determined by your offer. If she declines, the $10 shrinks to $6.67 and she then gets a

turn to make you an offer. Again, the bargain is complete if you accept and the division is made

according to the terms of her offer. If you decline her offer, the sum shrinks to $3.33 and you

get to make an offer. If your partner accepts, the division is made according to your offer, but if

she declines, the sum available shrinks to zero and you each get nothing.

If you are making the first offer in this bargaining game, what do you offer?

_________________

Situation 4

You are given $10 and told that you will be allowed to make one offer to split the sum with a

partner who will either accept or reject. Regardless of her answer, the game ends after her

move. If she accepts your offer, the $10 is split accordingly. If she rejects it, you both get

nothing.

What amount do you offer your partner in this game? ________________________

Situation 5

You are given $10 and told that you will be allowed to make only one offer to split the sum

with a partner. Your partner must either accept or reject. Regardless of her answer, the game

ends after her move. If she accepts your offer, the $10 is split accordingly. If she rejects your

offer, you get the full $10.

What amount do you offer your partner in this game? ________________________

The results from these bargaining games can be tabulated and given to students for

discussion in class. The first three situations allow discussion of the need to look ahead to later

rounds to consider the best possible first offer. (Variants with impatience can make these more

interesting.) In the last two games, the Nash equilibrium first offer is the same, although many

students do not offer the minimum (here 50 cents) in Situation 4 because of what they identify

as fear of rejection. This provides an additional opportunity to consider the validity of the Nash

equilibrium concept.